arXiv subjects

Barnaby Martin

Publications and source records attributed to Barnaby Martin.

At least 55 records · Page 3Linked to original sources

arXiv2018

Classification transfer for qualitative reasoning problems

We study formalisms for temporal and spatial reasoning in the modern context of Constraint Satisfaction Problems (CSPs). We show how questions on the complexity of their subclasses can be solved using existing results via the powerful use of primitive positive (pp) interpretations and pp-homotopy. We demonstrate the methodology by giving a full complexity classification of all constraint languages that are first-order definable in Allen's Interval Algebra and contain the basic relations (s) and (f). In the case of the Rectangle Algebra we answer in the affirmative the old open question as to whether ORD-Horn is a maximally tractable subset among the (disjunctive, binary) relations. We then generalise our results for the Rectangle Algebra to the r-dimensional Block Algebra.

cs.LO↗

arXiv2018

Disconnected Cuts in Claw-free Graphs

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several homomorphism and contraction problems, and fits in an extensive line of research on vertex cuts with additional properties. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms are known for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle $C_4$ led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The centerpiece of our result is a novel decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and expands the research line initiated by Chudnovsky and Seymour (JCTB 2007-2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further novel algorithmic results, namely Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

cs.DS↗

arXiv2017

Surjective H-Colouring over Reflexive Digraphs

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete. By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

cs.CC↗

arXiv2017

The complexity of quantified constraints using the algebraic formulation

Let A be an idempotent algebra on a finite domain. We combine results of Chen, Zhuk and Carvalho et al. to argue that if A satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if QCSP(Inv(A)) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture. We examine in closer detail the situation for domains of size three. Over any finite domain, the only type of PGP that can occur is switchability. Switchability was introduced by Chen as a generalisation of the already-known Collapsibility. For three-element domain algebras A that are Switchable, we prove that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible. The significance of this is that, for QCSP on finite structures (over three-element domain), all QCSP tractability explained by Switchability is already explained by Collapsibility. Finally, we present a three-element domain complexity classification vignette, using known as well as derived results.

cs.LO↗

arXiv2017

The packing chromatic number of the infinite square lattice is between 13 and 15

Using a SAT-solver on top of a partial previously-known solution we improve the upper bound of the packing chromatic number of the infinite square lattice from 17 to 15. We discuss the merits of SAT-solving for this kind of problem as well as compare the performance of different encodings. Further, we improve the lower bound from 12 to 13 again using a SAT-solver, demonstrating the versatility of this technology for our approach.

cs.DM↗

arXiv2016

On the Chen Conjecture regarding the complexity of QCSPs

Let A be an idempotent algebra on a finite domain. We combine results of Chen 2008 and Zhuk 2015 to argue that if Inv(A) satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture.

cs.LO↗

arXiv2016

Discrete Temporal Constraint Satisfaction Problems

A discrete temporal constraint satisfaction problem is a constraint satisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over $(\Bbb Z,<)$. Our main result says that every distance CSP is in Ptime or NP-complete, unless it can be formulated as a finite domain CSP in which case the computational complexity is not known in general.

math.LO↗

arXiv2016

Distance Constraint Satisfaction Problems

We study the complexity of constraint satisfaction problems for templates $Γ$ that are first-order definable in $(\Bbb Z; succ)$, the integers with the successor relation. Assuming a widely believed conjecture from finite domain constraint satisfaction (we require the tractability conjecture by Bulatov, Jeavons and Krokhin in the special case of transitive finite templates), we provide a full classification for the case that Gamma is locally finite (i.e., the Gaifman graph of $Γ$ has finite degree). We show that one of the following is true: The structure Gamma is homomorphically equivalent to a structure with a d-modular maximum or minimum polymorphism and $\mathrm{CSP}(Γ)$ can be solved in polynomial time, or $Γ$ is homomorphically equivalent to a finite transitive structure, or $\mathrm{CSP}(Γ)$ is NP-complete.

cs.CC↗

arXiv2015

Switchability and collapsibility of Gap Algebras

Let A be an idempotent algebra on a 3-element domain D that omits a G-set for a factor. Suppose A is not αβ-projective (for some alpha, beta subsets of D) and is not collapsible. It follows that A is switchable. We prove that, for every finite subset Delta of Inv(A), Pol(Delta) is collapsible. We also exhibit an algebra that is collapsible from a non-singleton source but is not collapsible from any singleton source.

cs.LO↗

arXiv2015

From complexity to algebra and back: digraph classes, collapsibility and the PGP

Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idempotent polymorphisms of certain digraph classes. Our first results are two algebraic dichotomy, even "gap", theorems. Building on and extending [Martin CP'11], we prove that partially reflexive paths bequeath a set of idempotent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP); or the exponentially generated powers property (EGP). Similarly, we build on [DaMM ICALP'14] to prove that semicomplete digraphs have the same property. These gap theorems are further motivated by new evidence that PGP could be the algebraic explanation that a QCSP is in NP even for unbounded alternation. Along the way we also effect a study of a concrete form of PGP known as collapsibility, tying together the algebraic and structural threads from [Chen Sicomp'08], and show that collapsibility is equivalent to its $Π_2$-restriction. We also give a decision procedure for $k$-collapsibility from a singleton source of a finite structure (a form of collapsibility which covers all known examples of PGP for finite structures). Finally, we present a new QCSP trichotomy result, for partially reflexive paths with constants. Without constants it is known these QCSPs are either in NL or Pspace-complete [Martin CP'11], but we prove that with constants they attain the three complexities NL, NP-complete and Pspace-complete.

cs.CC↗

arXiv2015

Constraint Satisfaction Problems around Skolem Arithmetic

We study interactions between Skolem Arithmetic and certain classes of Constraint Satisfaction Problems (CSPs). We revisit results of Glass er et al. in the context of CSPs and settle the major open question from that paper, finding a certain satisfaction problem on circuits to be decidable. This we prove using the decidability of Skolem Arithmetic. We continue by studying first-order expansions of Skolem Arithmetic without constants, (N;*), where * indicates multiplication, as CSPs. We find already here a rich landscape of problems with non-trivial instances that are in P as well as those that are NP-complete.

cs.CC↗

arXiv2014

The Computational Complexity of Disconnected Cut and 2K2-Partition

For a connected graph G=(V,E), a subset U of V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is NP-complete. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K2-partition, testing if a graph allows a vertex-surjective homomorphism to the reflexive 4-cycle and testing if a graph has a spanning subgraph that consists of at most two bicliques. Hence, as an immediate consequence, these three decision problems are NP-complete as well. This settles an open problem frequently posed in each of the four settings.

cs.CC↗

arXiv2014

Relativisation makes contradictions harder for Resolution

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d), as well as their tree-like versions, Res*(d). The contradictions we use are natural combinatorial principles: the Least number principle, LNP_n and an ordered variant thereof, the Induction principle, IP_n. LNP_n is known to be easy for Resolution. We prove that its relativisation is hard for Resolution, and more generally, the relativisation of LNP_n iterated d times provides a separation between Res(d) and Res(d+1). We prove the same result for the iterated relativisation of IP_n, where the tree-like variant Res*(d) is considered instead of Res(d). We go on to provide separations between the parameterized versions of Res(1) and Res(2). Here we are able again to use the relativisation of the LNP_n, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res*(1) and Res*(2). Here, the relativisation of IP_n will not work as it is, and so we make a vectorising amendment to it in order to address this shortcoming

cs.LO↗

arXiv2013

Constraint Satisfaction with Counting Quantifiers 2

We study constraint satisfaction problems (CSPs) in the presence of counting quantifiers $\exists^{\geq j}$, asserting the existence of $j$ distinct witnesses for the variable in question. As a continuation of our previous (CSR 2012) paper, we focus on the complexity of undirected graph templates. As our main contribution, we settle the two principal open questions proposed in (CSR 2012). Firstly, we complete the classification of clique templates by proving a full trichotomy for all possible combinations of counting quantifiers and clique sizes, placing each case either in P, NP-complete or Pspace-complete. This involves resolution of the cases in which we have the single quantifier $\exists^{\geq j}$ on the clique $K_{2j}$. Secondly, we confirm a conjecture from (CSR 2012), which proposes a full dichotomy for $\exists$ and $\exists^{\geq 2}$ on all finite undirected graphs. The main thrust of this second result is the solution of the complexity for the infinite path which we prove is a polynomial-time solvable problem. By adapting the algorithm for the infinite path we are then able to solve the problem for finite paths, and then trees and forests. Thus as a corollary to this work, combining with the other cases from (CSR 2012), we obtain a full dichotomy for $\exists$ and $\exists^{\geq 2}$ quantifiers on finite graphs, each such problem being either in P or NP-hard. Finally, we persevere with the work of (CSR 2012) in exploring cases in which there is dichotomy between P and Pspace-complete, in contrast with situations in which the intermediate NP-completeness may appear.

cs.LO↗

arXiv2013

On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction

The universal-algebraic approach has proved a powerful tool in the study of the complexity of CSPs. This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates, and relies on two facts. The first is that in finite or omega-categorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. The second is that every finite or omega-categorical structure is homomorphically equivalent to a core structure. In this paper, we present generalizations of these facts to infinite structures that are not necessarily omega-categorical. (This abstract has been severely curtailed by the space constraints of arXiv -- please read the full abstract in the article.) Finally, we present applications of our general results to the description and analysis of the complexity of CSPs. In particular, we give general hardness criteria based on the absence of polymorphisms that depend on more than one argument, and we present a polymorphism-based description of those CSPs that are first-order definable (and therefore can be solved in polynomial time).

cs.LO↗

arXiv2013

QCSP on partially reflexive cycles - the wavy line of tractability

We study the (non-uniform) quantified constraint satisfaction problem QCSP(H) as H ranges over partially reflexive cycles. We obtain a complexity-theoretic dichotomy: QCSP(H) is either in NL or is NP-hard. The separating conditions are somewhat esoteric hence the epithet "wavy line of tractability".

cs.CC↗

arXiv2012

On the complexity of the model checking problem

The model checking problem for various fragments of first-order logic has attracted much attention over the last two decades: in particular, for the primitive positive and the positive Horn fragments, which are better known as the constraint satisfaction problem and the quantified constraint satisfaction problem, respectively. These two fragments are in fact the only ones for which there is currently no known complexity classification. All other syntactic fragments can be easily classified, either directly or using Schaefer's dichotomy theorems for SAT and QSAT, with the exception of the positive equality free fragment. This outstanding fragment can also be classified and enjoys a tetrachotomy: according to the model, the corresponding model checking problem is either tractable, NP-complete, co-NP-complete or Pspace-complete. Moreover, the complexity drop is always witnessed by a generic solving algorithm which uses quantifier relativisation. Furthermore, its complexity is characterised by algebraic means: the presence or absence of specific surjective hyper-operations among those that preserve the model characterise the complexity.

cs.LO↗

arXiv2012

Containment, Equivalence and Coreness from CSP to QCSP and beyond

The constraint satisfaction problem (CSP) and its quantified extensions, whether without (QCSP) or with disjunction (QCSP_or), correspond naturally to the model checking problem for three increasingly stronger fragments of positive first-order logic. Their complexity is often studied when parameterised by a fixed model, the so-called template. It is a natural question to ask when two templates are equivalent, or more generally when one "contain" another, in the sense that a satisfied instance of the first will be necessarily satisfied in the second. One can also ask for a smallest possible equivalent template: this is known as the core for CSP. We recall and extend previous results on containment, equivalence and "coreness" for QCSP_or before initiating a preliminary study of cores for QCSP which we characterise for certain structures and which turns out to be more elusive.

cs.LO↗